qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,360,515 | <p>EDIT: Added some clarifications.
EDIT 2: More clarifications and 1 more example.</p>
<p>Here's the basis of the problem:</p>
<p>I have 2 numbers, (they are integers that can be separated by 0.5 or 1, i.e. 1; 1.5; 2; 2.5; ...) let's say as an example these numbers are 22.5 and 10.</p>
<p>Now you must turn the bigg... | Jonathan | 706,053 | <p>In your formula for OLS solution <span class="math-container">$H$</span>, <span class="math-container">$H=X(X^TX)^{-1}X^T$</span>, you are assuming that <span class="math-container">$X$</span> has a full column-rank, which implies that <span class="math-container">$R^T R$</span> is invertible, i.e. <span class="math... |
856,814 | <p>For which x does the function $f(x) = x^3-6x^2-5x+5$ assume its maximum value on the interval $[-5,5]$?</p>
<p>The critical points for this function are $\frac{12 + \sqrt{204}}{6}$ and $\frac{12 - \sqrt{204}}{6}$.
The end points are -5 and 5.</p>
<p>The maximum value is solved to be for $\frac{12 - \sqrt{204}}{6}\... | Pearl Philip | 156,880 | <p>The probability that you will hit the target always remains $0.3$ each time you try. This is because each attempt is independent and is unaffected by successes and failures from previous attempts. The total probability of hitting the target all of the $15$ times you try will be </p>
<p>$P=(0.3)^{15}= 0.000000014348... |
31,113 | <p>Zagier has a very short proof (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1041893" rel="noreferrer">MR1041893</a>, <a href="https://www.jstor.org/stable/2323918" rel="noreferrer">JSTOR</a>) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an inv... | Moritz Firsching | 39,495 | <p>Let me answer your question "where do these involution come from" with an elementary geometric explanation. You can skip ahead to the pictures, which are somewhat self-explanatory, I hope. </p>
<p>The elements in the the $S$, i.e. triplets $(x,y,z)$ such that $x^2+4yz=p$, can be visualized as a square of side lengt... |
1,989,764 | <blockquote>
<p>Let $f,g$ be continuous functions on $[a,b]$ and suppose $g(x) \geq 0
$ for all $x \in [a,b]$. Prove there exists $\xi \in [a,b]$ such that </p>
<p>$$ \int\limits_a^b f(x) g(x) \mathrm{d} x = f( \xi ) \int\limits_a^b
g(x) \mathrm{d} x $$</p>
</blockquote>
<p>My attempt:</p>
<p>Mean value theo... | Ernesto Iglesias | 378,968 | <p>Matrix operations are linear and translation is not. I can't understand why you use moving matrix (go to 3-d spaces for something like that).</p>
<p>Anyway if you have to do something like that: The Doug M. answer is correct.</p>
<p>$\begin{bmatrix}x'\\ y'\\ 1\end{bmatrix}=
\begin{bmatrix}1&0&2\\ 0&1&a... |
155,143 | <p>Let $X$ be a Banach space an let $(e_n)_{n=1}^{\infty}$ be a Schauder basis of $X$. If we denote the sequence of the natural projections associated with $(e_n)_{n=1}^{\infty}$ by $(S_n)_{n=1}^{\infty}$ then by Uniform Boundedness principle we can observe that $$\sup_n||S_n||< \infty. $$</p>
<p>And the number $K=... | Michael Renardy | 12,120 | <p>A basis with basis constant 1 is called monotone. You can always find an equivalent norm such that the basis constant is 1. It is very simple: Simply define the new norm as $|||x|||=\sup_n \|S_nx\|$. </p>
<p>A more interesting question is whether this new norm satisfies any "desirable" properties, such as uniform c... |
3,137,775 | <p>Let <span class="math-container">$Z$</span> be the matrix </p>
<p><span class="math-container">$$Z =
\begin{pmatrix} 1 & 2\\ 3 & 4 \end{pmatrix}$$</span></p>
<p>And the set <span class="math-container">$\text{Commu}(Z) = \{B ∈ M_{2\times2} | ZB = BZ\}$</span>.</p>
<p>What is the set <span class="math-cont... | GeorgSaliba | 142,772 | <p>For completeness, here is yet another way of solving it:</p>
<p>Set <span class="math-container">$x^2-1=t^2x^2$</span> then <span class="math-container">$$x=\frac{1}{\sqrt{1-t^2}} \qquad t=\frac{\sqrt{x^2-1}}{x}$$</span> and <span class="math-container">$$dx=\frac {tdt}{(1-t^2)^{3/2}}$$</span></p>
<p>The integral ... |
4,205,525 | <p>I have to prove by mathematical induction that <span class="math-container">$L = R$</span></p>
<p>Question:</p>
<p><a href="https://i.stack.imgur.com/3UYsv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3UYsv.png" alt="enter image description here" /></a></p>
<p>What I've tried so far:</p>
<p><a ... | kirigaya | 888,007 | <p>We will proceed by mathematical induction. if <span class="math-container">$n=1$</span> then we have <span class="math-container">$5(1)-1=4$</span> and <span class="math-container">$\frac{1(5*1+3)}{2}=4$</span>. Therefore, the proposition holds for <span class="math-container">$n=1$</span>. Suppose that <span class=... |
2,079,496 | <p>Let $D$ be a finite integral domain with unity and let $x \in D$. Is it true that then $x^{-1} = x^n$ for some $n \in N$, and thus every element has an inverse. This was my solution to proving that $D$ is also a field. I am thinking of the analogue proof in groups (See Grillet's abstract algebra corollary 2.4) and s... | Asinomás | 33,907 | <p>An alternative proof: The invertible elements in any ring form what is called the multiplicative group under multiplication. Since we are now in a finite group we have that $a^{-1}=a^k$ for some $k$ (by your previous result for groups).</p>
|
973,771 | <p>I am trying to find </p>
<p>$$ \lim_{x\to 0} \frac{e^{\alpha x} -e^{\beta x}}{\sin(\alpha x) + \sin(\beta x) }$$</p>
<p>but dont have a clue where to start, could someone give me a hint please ?</p>
| ajotatxe | 132,456 | <p>The expression makes sense only if $\alpha\neq-\beta$, and if it is the case, you can simply apply l'Hopital's rule (if you want only a hint, stop reading now):</p>
<p>$$\begin{align}
\lim_{x\to 0}\frac{e^{\alpha x}-e^{\beta x}}{\sin(\alpha x)+\sin(\beta x)}&=\lim_{x\to0}\frac{\alpha e^{\alpha x}-\beta e^{\beta... |
2,980,201 | <p>I'm trying to prove that:</p>
<p><span class="math-container">$$
\sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty
$$</span></p>
<p>where <span class="math-container">$|r_1|, |r_2| < 1$</span> and <span class="math-container">$c_1, c_2$</span> are some arbitraty real numbers (<span class="math-container">$r_... | Siong Thye Goh | 306,553 | <p><span class="math-container">$$\sum_{j=0}^\infty |c_1r_1^j + c_2r_2^j| \le \sum_{j=0}^\infty |c_1||r_1|^j + \sum_{j=0}^\infty|c_2||r_2|^j$$</span></p>
<p>By comparison test, it converges.</p>
|
43,043 | <p>I'm working through Carl Bender's <a href="http://youtu.be/LYNOGk3ZjFM">Mathematical Physics lectures</a> on YouTube (which are great fun), and I'd like <em>Mathematica</em>'s help solving terms in the perturbation series. It would be convenient if expressions like</p>
<pre><code>SeriesCoefficient[Sum[a[n] b^n, {n,... | Ian | 712 | <p>The (accepted) answer given by episanty gave me another idea I want to share. While <code>SeriesCoefficient</code> leaves the infinite sum untouched, <code>Derivative</code> does not, so we can get the coefficients manually. The issue episanty points out is that the lower limit of the sum errors out at $0^0$. We can... |
2,164,338 | <p>I am stuck up with this simple problem related to averages. Please help me out in explaining the complete solution to this problem.</p>
<blockquote>
<p>Consider a class of <span class="math-container">$40$</span> students whose average weight is <span class="math-container">$40\ \mathrm{kg}$</span>. <span class="... | quasi | 400,434 | <p>After the join, the new sum is</p>
<p>\begin{align*}
&(40)(40) + mn\\[6pt]
=\; &1600 + mn\\[6pt]
=\; &1600 + m(50-m)
\end{align*}</p>
<p>and the new average is</p>
<p>$$f(m) = \frac{1600 + m(50-m)}{40 + m}$$</p>
<p>We want to maximize $f(m)$, for positive integer values of $m$. Taking the derivative... |
980,470 | <p>How to find the continued fraction of $\sqrt{n}$, for an integer $n$?</p>
<p>I saw a site where they explained it, but it required a calculator. Is it possible to do it without a calculator? </p>
| robjohn | 13,854 | <p>Let's try, for example, $\sqrt5$:</p>
<p>Since that $2\lt\sqrt5\lt3$, the first term is $2$. Subtract $2$ and invert:
$$
\frac1{\sqrt5-2}=\sqrt5+2
$$
Since $4\le\sqrt5+2\le5$, the next term is $4$. Subtract $4$ and invert:
$$
\frac1{\sqrt5+2-4}=\sqrt5+2
$$
We are at the same point as the previous line. Thus, the co... |
30,339 | <p>I need to construct a vector similar to:</p>
<pre><code>v[x_]:={0, 0, x, 0, 0, 2*x, 0, 0, 3*x, ....., 0, 0, n*x};
</code></pre>
<p>where <code>n=10^9</code>. How can I make such a vector wisely?</p>
| Kuba | 5,478 | <p>There are many ways, for example:</p>
<pre><code> n = 5;
l = SparseArray[i_?(Divisible[#, 3] &) -> i/3 x, {3n}, 0]
</code></pre>
<blockquote>
<pre><code>SparseArray[<5>,{15}]
</code></pre>
</blockquote>
<pre><code> List @@ l
</code></pre>
<blockquote>
<pre><code>{0, 0, x, 0, 0, 2 x, 0, 0, 3 x, 0, ... |
864,823 | <p>given the following horses, each with its chance of winning:</p>
<p>Horse 1 -> 0.29<br>
Horse 2 -> 0.34<br>
Horse 3 -> 0.11<br>
Horse 4 -> 0.07<br>
Horse 5 -> 0.14<br>
Horse 6 -> 0.05<br>
Sum -> 1</p>
<p>At the moment, in order to calculate the straight forecast of 1-4, that is the probability that horse 1 wins th... | Community | -1 | <p>From your solution, I thought to consider the change of variable $f(x) = 1 + g(x)$: your functional equation then becomes</p>
<p>$$ g(x) g(-x) = 1 $$</p>
<p>and now the entire solution space becomes obvious: you can pick $g(x)$ to be <em>any</em> function on the positive reals that is nowhere zero, and then the va... |
3,060,789 | <blockquote>
<p>Prove that a group with <span class="math-container">$3$</span> elements is cyclic.</p>
</blockquote>
<p>I tried the case where <span class="math-container">$G=\{e,a,b\}
$</span>
and I kept trying multiplication
and finally I found that
<span class="math-container">$a^2$</span> must equal to <span ... | Coupeau | 626,733 | <p>According to <a href="https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)" rel="nofollow noreferrer">Lagrange's theorem</a>, a group with 3 elements can have only the trivial subgroups. Because 3 is prime only 1 and 3 divide it. So, only possible subgroups are <span class="math-container">$\{e\}$</span... |
1,780,751 | <p>If $$I=\int\sin 2x\, \mathrm{d}x$$
since $$(\sin^2x)'=2\cos x\sin x=\sin 2x$$ then $$I=\sin^2x+C$$</p>
<p>However using substitution
$$I=\int \frac{\sin2x}{-2\sin 2x}\,\mathrm{d}u,~~~~\textrm{s.t}~~~ u=\cos2x$$
giving $$I=-\frac{1}{2}\int\,\mathrm{d}u=-\frac{1}{2}\cos2x+C=\sin^2x-\frac{1}{2}+C$$</p>
<p>I have chec... | Ian Miller | 278,461 | <p>Both answers are correct. As you know when you integrate you always need to add a constant. In your two answers the constants are expressed different. In the first you have labelled it $C$ and the second you have labelled it $C-\frac{1}{2}$.</p>
<p>In otherwords your two constants different by a factor of $\frac12$... |
340,485 | <p>If $f^{'}(3) = 2$ then find</p>
<p>$$\lim_{h \rightarrow 0}\left(\frac{f(3+h^2)-f(3-h^2)}{2h^2} \right)$$</p>
| jock214 | 68,393 | <p>When $x=2$ let $f=(3-h)^2=-2h$, use this to simplify the original equastion</p>
|
3,549,582 | <blockquote>
<p>Let V and w be finite-dimensional vector spaces with ordered bases <span class="math-container">$\beta={v_1,v_2,...,v_n}$</span> and <span class="math-container">$\gamma={w_1,w_2,...,w_m}$</span>. Let <span class="math-container">$T:V \rightarrow$</span> be linear. Then for each j, <span class="math-c... | angryavian | 43,949 | <p>The only way he fishes for more than two hours is if he catches zero fish in the first two hours. But in this situation he will only go home with one fish (the first fish he catches after the first two hours, after which he immediately goes home). So the only way he catches more than two fish is if he gets those two... |
2,748,162 | <blockquote>
<p>An elevator with capacity for 10 people is designed to support a maximum load of 750kg.If the weights of people that use the elevator are distribuited with mean 70kg and standard deviation 8kg. Which is the probability that a group of 10 people exceed the maximum load ? </p>
</blockquote>
<p>Attempt... | Quoka | 410,062 | <p>If you are assuming a normal distribution then $X_i \sim \mathcal{N}(70, 8^2)$. Therefore,
$$
X=\sum_{i=1}^{10} X_i \sim \mathcal{N}(700, 640)
$$
Finally,
$$
P(X\leq 750) = P\left(\frac{X-700}{\sqrt{640}} \leq \frac{50}{\sqrt{640}}\right) = P\left(Z \leq \frac{50}{\sqrt{640}}\right)
$$
where $Z$ is the Standard norm... |
2,748,162 | <blockquote>
<p>An elevator with capacity for 10 people is designed to support a maximum load of 750kg.If the weights of people that use the elevator are distribuited with mean 70kg and standard deviation 8kg. Which is the probability that a group of 10 people exceed the maximum load ? </p>
</blockquote>
<p>Attempt... | Hagen von Eitzen | 39,174 | <p>When adding independent normally distributed random vars, their sum is normally distributed where the mean is the sum of the means and the variance (square of standard deviation) is the sum of the variances.</p>
<p>So here we get that the mean is $10\cdot 70\,\rm{kg}$ and the variance is $10\cdot 64\,\rm{kg}^2$</p>... |
2,748,162 | <blockquote>
<p>An elevator with capacity for 10 people is designed to support a maximum load of 750kg.If the weights of people that use the elevator are distribuited with mean 70kg and standard deviation 8kg. Which is the probability that a group of 10 people exceed the maximum load ? </p>
</blockquote>
<p>Attempt... | Phil H | 554,494 | <p>Figure out the Z score for a mean of 75 kg in a normal distribution with a mean of 70 kg and SD of 8 kg. Z = (75 - 70)/(8/sqrt 10). From a Z table read off the p value and subtract it from 1, which is then the area to right of the Z score as the probability of exceeding the max weight. You can also do this on a TI-8... |
368,745 | <p>Here $\mu$ is a probability measure. Another similar question is: is it a $\sigma$-compact space? Thank you in advance!</p>
| tmh | 36,102 | <p>Let's assume $\mu$ is not a finite linear combination of point masses, in which case the answer is trivially yes.</p>
<p>Otherwise, the answer is no.</p>
<p>$L^\infty(\mu)$ is a Banach space, and we have that, in general for Banach spaces $B$,</p>
<p>$B$ locally compact $\iff$ $B$ is finite dimensional $\iff$ $B$... |
4,267,359 | <p><strong>The problem:</strong></p>
<p>If <span class="math-container">$f(x)=\begin{cases}\frac{\sin^2(ax)}{x^2}&x\neq0\\ 1&x=0\end{cases}$</span>, show that <span class="math-container">$f(x)$</span> is discontinuous at <span class="math-container">$x=0$</span>. If <span class="math-container">$f(x)$</span> i... | José Carlos Santos | 446,262 | <p>Actually, <span class="math-container">$\displaystyle\lim_{x\to0}\frac{\sin^2(ax)}{x^2}=a^2$</span>. Therefore, <span class="math-container">$f$</span> is continuous at <span class="math-container">$0$</span> if and only if <span class="math-container">$a=\pm1$</span>.</p>
|
257,279 | <p>The <strong>Proper Forcing Axiom</strong> kills CH in a particularly specific way: it implies that $2^{\aleph_0}=\aleph_2$. However, its impact on the continuum function above $\aleph_0$ is much less clear. It is known, for example, that it implies the <em>Singular Cardinal Hypothesis</em>, SCH.</p>
<p>In <a href="... | Pietro Majer | 6,101 | <p>I think there is not such an $f:S\to S$. Consider the sequence $x^t\in S$ continuously depending on $t\in[0,1]$, such that $x_0^t=-t$ and $x_n^t=1/n $ for all $n\ge1$. Since $f(x^1)$ is dense, for some index, say $17$, we have $-1 <f_{17}(x^1)<0$. Therefore, for $t=1$, we have $$-t=x^t_0<f_{17}(x^t)<x... |
436,071 | <blockquote>
<p>Calculate the double integral $\iint_D {(1+x^2 + y^2)ln(1+x^2+y^2)dxdy} $ where $D = \{(x,y) \in \mathbb R^2 | \frac{x}{\sqrt3} \leq y \leq x , x^2 + y^2 \leq 4\}$.</p>
</blockquote>
<p>I heard there is a way called Polar Coordinates but the more I looked and read about it the more I did not understa... | Ron Gordon | 53,268 | <p>Taking @Daniel Fisher's comment to its logical conclusion, the lines imply a circular sector in which $\theta \in [\pi/6,\pi/4]$. Thus the area integral is</p>
<p>$$\int_{\pi/6}^{\pi/4} d\theta \, \int_0^2 dr \, r \, (1+r^2) \, \ln{(1+r^2)}$$</p>
|
4,039,241 | <p>Let <span class="math-container">$\mathcal{C}([-1, 1]) = \{f: [-1, 1] \to \mathbb{R}\mid f \text{ is continuous}\}$</span>, <span class="math-container">$\mathcal{C}^{1}([-1, 1]) = \{f: [-1, 1] \to \mathbb{R}\mid f' \text{ is continuous}\}$</span>, where both sets are equipped with the supremum norm <span class="mat... | StratosFair | 857,384 | <p>No, this is not true. If, for instance <span class="math-container">$X \sim \text{Binomial}(n,p)$</span> with <span class="math-container">$n = 10$</span> and <span class="math-container">$p=0.55$</span> then <span class="math-container">$\mathbb E[X] = 5.5$</span> and <span class="math-container">${\mathbb{E}\left[... |
173,187 | <p>Does anyone how to solve this matrix equation?
$$ PXQ^T+P^TXQ=A$$
where all matrices are real and square. Can you provide me with some guidelines?
Thanx</p>
| Federico Poloni | 1,898 | <p>If you are solving this numerically on a computer, you should use a variant of the Bartels-Stewart method; see for instance <a href="http://dl.acm.org/citation.cfm?id=146929" rel="nofollow">http://dl.acm.org/citation.cfm?id=146929</a> (Gardiner, Laub, Amato, Moler).</p>
<p>This costs $O(n^3)$ operations rather than... |
2,279,581 | <blockquote>
<p>Let $a,b$ be primes. Find all primes such that $a^2+b^2=9ab-13$. </p>
</blockquote>
<p>Whatever I've done is from parity checking. But I can't proceed with the case when both $a,b$ are odd primes.
I tried with some modulo-chasing but couldn't complete.</p>
| rtybase | 22,583 | <p><strong>Not a complete answer</strong>, just a few restrictions...</p>
<p><strong>Proposition 1.</strong> For any prime $p>3 \Rightarrow 3 \mid p^2-1$ (from <a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow noreferrer">LFT</a>) and $8 \mid p^2-1$ (from $(2k+1)^2 \equiv 1 \pmod{8}$).... |
285,965 | <p>I was told to assume $f(x)$ is a polynomial with degree $d\geq 1$ with integer coefficients and positive leading coefficient.</p>
<p>(i) I need to show that there are infinitely many $x$ such that $f(x)$ isn't prime.</p>
<p>(ii) I also need to show that if $f(x_0) = m$, where $m>0$, then $f(x) \equiv 0 \pmod m$... | Hagen von Eitzen | 39,174 | <p>For (ii) note that $(x_0+km)^n=x_0^n+{n\choose 1}x_0^{n-1}m+{n\choose 2}x_0^{n-2}m^2+\ldots +m^n = x_0^n+m\cdot(\ldots)$, therefore in general $f(x_0+km)\equiv f(x_0)\pmod m$.</p>
<p>There are at most finitely many $x$ with $f(x)=0$, so we can pick $x_0$ such that $m:=f(x_0)\ne 0$.
Among the infinitely many $x=x_0+... |
3,807,327 | <p>From <a href="https://www.amazon.de/Introduction-Mathematical-Statistics-Global-Robert/dp/1292264764/ref=sr_1_1?dchild=1&keywords=Introduction%20to%20Mathematical%20Statistics&qid=1598710468&sr=8-1" rel="nofollow noreferrer">Introduction to Mathematical Statistics</a> Page 198</p>
<p>Let (X,Y) follow a b... | Khashayar Baghizadeh | 798,891 | <p>Today I solved this problem again and I noticed that</p>
<p>"Let <span class="math-container">$u$</span> be an arbitrary element of <span class="math-container">$U$</span> and so <span class="math-container">$\forall b\in B(bRu)$</span>. Since <span class="math-container">$x\in L$</span>, <span class="math-cont... |
282,382 | <p>Let $s\text{Ring}$ denote the category of simplicial commutative rings.
We endow it with the model structure defined by declaring that fibrations, trivial fibrations and weak equivalences are, respectively, those maps inducing fibrations, trivial fibrations and weak equivalences on underlying simplicial sets.</p>
<... | Community | -1 | <p>This is just a small complement to Charles' answer.
The category of simplicial abelian groups is finitely generated (see Hovey's book $\S$7, for instance) and domains and codomains of generating cofibrations, respectively trivial cofibrations, are small relative to the whole category, since they are finitely present... |
3,030,894 | <p>Given that <span class="math-container">$f(x)=\sin x+\tan x$</span> on <span class="math-container">$(-\pi/2,\pi/2)$</span>. Show that it has an inverse function. Find the derivative of the inverse of <span class="math-container">$f(x)$</span> at <span class="math-container">$x=0$</span>.</p>
<p>What is the approp... | BigbearZzz | 231,327 | <p>An alternative way to show that <span class="math-container">$f$</span> is continuous at <span class="math-container">$x=0$</span> is to use the sandwich theorem with
<span class="math-container">$$
-|x| \le f(x) \le |x|,
$$</span>
where <span class="math-container">$\pm|x|\to 0 $</span> as <span class="math-contain... |
3,030,894 | <p>Given that <span class="math-container">$f(x)=\sin x+\tan x$</span> on <span class="math-container">$(-\pi/2,\pi/2)$</span>. Show that it has an inverse function. Find the derivative of the inverse of <span class="math-container">$f(x)$</span> at <span class="math-container">$x=0$</span>.</p>
<p>What is the approp... | Michael Hoppe | 93,935 | <p>It'v very easy to show that for any bounded function <span class="math-container">$g$</span>, defined on a neighbourhood of <span class="math-container">$0$</span>, the function <span class="math-container">$f(x)=x\cdot g(x)$</span> is continuous at <span class="math-container">$x=0$</span>. </p>
<p>For the differe... |
79,991 | <p>I'm attempting to MapThread a function of two lists that requires the index of the list values. For example,</p>
<pre><code>MapThread[#1*i+#2*j&,{{a,b,c},{e,f,g}}]
</code></pre>
<p>Where #1 represents the value from list 1, #2 the value from list 2, i the index of list 1, and j the index of list 2. The expecte... | m_goldberg | 3,066 | <p>Here is definition for <code>indexedMapThread</code> that works for any number of lists so long as they are all equal in length.</p>
<pre><code>indexedMapThread[args : {_List ..}] :=
Module[{sizes = Length /@ args, scalars},
If[Not[Equal @@ sizes], Return[$Failed]];
scalars = Range@sizes[[1]];
Expand @ Flatt... |
79,991 | <p>I'm attempting to MapThread a function of two lists that requires the index of the list values. For example,</p>
<pre><code>MapThread[#1*i+#2*j&,{{a,b,c},{e,f,g}}]
</code></pre>
<p>Where #1 represents the value from list 1, #2 the value from list 2, i the index of list 1, and j the index of list 2. The expecte... | Mr.Wizard | 121 | <p>Since it would seem that your index values <em>i</em> and <em>j</em> will always be the same you need only to <a href="http://reference.wolfram.com/language/ref/Transpose.html" rel="nofollow"><code>Transpose</code></a> your input and use <a href="http://reference.wolfram.com/language/ref/MapIndexed.html" rel="nofoll... |
1,609,845 | <p><strong>Problem</strong>: Let $\alpha, \beta$ be non-negative numbers.
Suppose the number of strictly increasing sequences $a_0, a_1, a_2 \cdots a_{2014}$ satisfying $0 \leq 3m$ is $2^{\alpha}(2\beta+1)$.
Find $\alpha$.</p>
<p>My attempt:</p>
<p>I tried to find an explicit form for the $nth$ of the sequence using... | Start wearing purple | 73,025 | <p>This can be done by a kind of residue calculation. First of all by parity the integral is $\frac12 \int_{-\infty}^{\infty}$. Considering this as a contour integral in the $x$-plane, we would like to pull the contour to $i\infty$. There will be two obstacles:</p>
<ul>
<li><p>an infinite number of poles of $\psi(x)$ ... |
3,873,630 | <p>Hi Guys can some one help with this question?</p>
<p>Let <span class="math-container">$\mathcal{H}$</span> be a Hilbert space and F a linear bounded functional in <span class="math-container">$\mathcal{H}^∗$</span>
such that <span class="math-container">$F \neq 0$</span>. Prove that dim(kerF)<span class="math-contai... | M477 | 839,811 | <p>I have come up with a solution and it seems to be correct.</p>
<p>If you hope to apply the R14, then you will need to introduce <span class="math-container">$\overline{L}(a,\textbf{a})$</span> which is, by usual definition, <span class="math-container">$\langle \langle F(0,\textbf{a}),G(0,\textbf{a})\rangle,...,\lan... |
53,373 | <p>Referring to Lang's Algebra p.137 P1, an $A$-module $P$ is projective if the following property holds: given $A$-homomorphism
$f:P \longrightarrow M^{''}$ and surjective $A$-homomorphism $g:M \longrightarrow M^{''}$, then
there exists $A$-homomorphism $h:P \longrightarrow M$ such that $f = g \circ ... | Pete L. Clark | 299 | <p>Theorem III.4.1 on p. 135 of Lang's <em>Algebra</em> says:</p>
<blockquote>
<p>Let <span class="math-container">$A$</span> be a ring and <span class="math-container">$M$</span> a module over <span class="math-container">$A$</span>. Let <span class="math-container">$I$</span> be a nonempty set, and let <span class="... |
3,194,279 | <p>I'd like to find a smooth function to approximate <span class="math-container">$$f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$$</span>
This function should be differentiable everywhere. Thanks.</p>
| Varun Vejalla | 595,055 | <p>From <a href="https://www.quora.com/What-smooth-approximations-to-the-ReLu-function-are-available-in-TensorFlow" rel="nofollow noreferrer">https://www.quora.com/What-smooth-approximations-to-the-ReLu-function-are-available-in-TensorFlow</a>, an approximation is <span class="math-container">$\frac{1}{t}$</span>ln<spa... |
4,464,680 | <p>While trying to solve the complexity of my program I came across the the following summation:</p>
<p><span class="math-container">$$\sum_{k = 1}^n (k \log k)\binom{n}{k}$$</span></p>
<p>Could you please provide a solution to this sum. If it is difficult to obtain the exact solution, could you please provide an asymp... | Sangchul Lee | 9,340 | <p>User <strong>@runway44</strong> already showed that</p>
<p><span class="math-container">$$ \sum_{k=1}^{n} (k \log k) \binom{n}{k} \leq n2^{n-1}\log n.$$</span></p>
<p>To show that this upper bound is "sharp", note that the function</p>
<p><span class="math-container">$$ f(x) = \begin{cases}
x \log x, &... |
4,455,969 | <p>The following question is part of a chemistry problem that came in the Dhaka University admission exam 2013-14.</p>
<hr />
<blockquote>
<p>What is <span class="math-container">$-\log\left({\sqrt{1.8\times 10^{-5}\times 0.1}}\right)$</span>? <br> <br>
(a) 2.672 <br> <br>
(b) 2.772 <br> <br>
(c) 2.872 <br> <br>
(d) 2.... | Claude Leibovici | 82,404 | <p>Working with whole numbers
<span class="math-container">$$1.8\times 10^{-5}\times 0.1= \frac 9 5 \times 10^{-6}$$</span>
<span class="math-container">$$\sqrt{1.8\times 10^{-5}\times 0.1}=\frac{3}{ \sqrt{5}}\times 10^{-3}$$</span>
<span class="math-container">$$\frac{3}{ \sqrt{5}}=\frac{3}{ \sqrt{4+1}}=\frac 3 2 \tim... |
746,557 | <p>An exercise from Introductory Combinatorics by Richard A.Brualdi:</p>
<blockquote>
<p>A roller coaster has five cars, each containing four seats, two in front and two
in back. There are 20 people ready for a ride. In how many ways can the ride
begin? What if a certain two people want to sit in different ca... | 0xcaff | 232,717 | <p>There is a much simpler solution to this problem.</p>
<p>There are 4 distinguishable seats in each car (one front left seat, one front right seat, one back left seat, one back right seat). There are 5 distinguishable cars. There are 20 distinguishable seats total. The first person has 20 choices, the second person ... |
1,517,291 | <p>The question is : <a href="https://i.stack.imgur.com/sG4uW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sG4uW.png" alt="enter image description here"></a></p>
<p>I am stuck here :</p>
<p>$$| \sqrt{x+1} - \sqrt{x} -1 | / \sqrt{x} + 1$$</p>
<p>i know that the numerator is negetive so i must ch... | SchrodingersCat | 278,967 | <p>The solution goes as follows:
$$\lim_{x\to\infty}\frac{\sqrt{x+1}}{\sqrt{x}+1}$$
$$=\lim_{\frac{1}{x}\to 0}\frac{\sqrt{1+\frac{1}{x}}}{1+\frac{1}{\sqrt{x}}}$$
$$=\frac{1+0}{1+0}$$
$$=1$$</p>
|
195,413 | <p>I have tried to write a code to check whether a line is parallel to the plane in Mathematica.
Using that a plane has the normal vector of <span class="math-container">$\vec{n}=(a,b,c)$</span> and a line has a direction vector <span class="math-container">$\vec{L}=(l,m,n),$</span> the condition is <span class="math-c... | nufaie | 63,341 | <pre><code>{a, b, c, d} = {1, 2, 3, 4};
plane = ImplicitRegion[a x + b y + c z + d == 0, {x, y, z}];
{x0, y0, z0} = {2, 1, 4};
{l, m, n} = {3, 1, 9};
line = ImplicitRegion[(x - x0)/l == (y - y0)/m == (z - z0)/n, {x, y,
z}];
If[(a*l + b*m + c*n == 0), Return["parallel"],
Return["not parallel"]];
</code></pre>
<p... |
2,074,785 | <p>I am looking for a formula for the volume $V_i(r,L)$ of the intersection between a sphere of radius $r$ and a cube of edge $L$ with the same center.</p>
<p>For $r<\frac L 2$, $V_i$ is simply equal to the volume of the sphere:</p>
<p>$$V_i(r,L) = \frac 4 3 \pi r^3 \ \ \ \ r<\frac L 2$$</p>
<p>For $r>\frac... | John Hughes | 114,036 | <p>Since you speak of an "easy way" to compute the volume in this third case, I can only say this: I don't think that there <em>is</em> an easy way. My thoughts and (modest) initial efforts to solve this follow, but they don't finish with an actual formula, so read on only for inspiration, not a solution. </p>
<p>Firs... |
3,748,284 | <p>I have already tried Wolfram Alpha (not pro) and I don't know whether I can access MATLAB for free, any software than downloaded for free and is easy to use will work. I need it for computing complex Hermite polynomials and solving differential equations of order more than 3.</p>
| Chris | 745,888 | <p><a href="https://perso.crans.org/besson/matlab-clones.en.html" rel="nofollow noreferrer">Here</a> are some free alternatives to matlab, although from my experience none are going to really be quite up to par.</p>
|
1,852,826 | <p>Let $P_n$ be the $(n+1) \times (n+1)$ matrix that contains the numbers of Pascal's triangle in the upper triangle. For example in the case of $n=3$
$$
P_3 =
\begin{pmatrix}
1 & 1 & 1 & 1 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 3 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}
$$
or in gene... | Jyrki Lahtonen | 11,619 | <p>In my opinion this is a bit simpler to prove, if we interpret the matrix <span class="math-container">$P_n$</span> as a linear transformation.</p>
<p>Consider the space <span class="math-container">$V_n$</span> of polynomials of degree <span class="math-container">$\le n$</span> (over, say <span class="math-contain... |
1,509,205 | <p>If $n$ is a positive Integer,then $43^{5n}-21^{5n}$ is always divisible by</p>
<p>options:</p>
<p>a)$11\quad$ b)$18\quad$ c)$25\quad$ d)$64$</p>
<p>MyApproach:</p>
<p>I am not able to think how to solve this problem but.</p>
<p>I did $43^{5n}-21^{5n}=a^n-b^n$ where a=$43^5$ and b=$21^5$</p>
<p>So, $a^n-b^... | mvw | 86,776 | <p>Assumed problem: Given the setting
\begin{align}
A = \text{tridiag}(a, b, c) \in \mathbb{R}^{n\times n} & \\
a \in \mathbb{R}^n & \quad \text{(main diagonal)} \\
b, c \in \mathbb{R}^{n-1} & \quad \text{(upper and lower side diagonal)} \\
x \in \mathbb{R}^n &
\end{align}
you want to calculate $y = Ax... |
1,937,351 | <p>$$\sum_{n=1}^\infty (-1)^n$$
Is this mathematical expression zero or undefined?
I think it looks like zero but i can't explain the reason mathematically.
In addition, $\infty - \infty$ is undefined afaik?</p>
| basket | 294,706 | <p>It is undefined. The sequence of partial sums is $-1, 0, -1, 0...$ which, while bounded, does not converge.</p>
|
2,324,718 | <p>To find the inverse of a function, you switch the spots of $x$ and $y$ in the function and solve for the new $y$ output. For example:</p>
<p>$y = 3 + \sin\ x$</p>
<ul>
<li>Switch positions of $x$ and $y$</li>
</ul>
<p>$x = 3 + \sin\ y$</p>
<ul>
<li>Solve for this new $y$</li>
</ul>
<p>$y = \sin^{-1}(x-3)$</p>
... | Saketh Malyala | 250,220 | <p>Consider this magic box.</p>
<p>$X$ => [ $F$ ] ==> $Y$</p>
<p>Now, by solving for $X$ in terms of $Y$, we have effectively found</p>
<p>$X$ <== [$F^{-1}$] <== $Y$.</p>
<p>Now, we just switch the names of $X$ and $Y$, because convention tell us to always express $Y$ in terms of $X$, even in inverse function... |
3,100,381 | <p>I am trying to prove this, but I don't know if I'm on the right track.</p>
<p>So far I've said <span class="math-container">$d=(a,c)=1$</span>, therefore, <span class="math-container">$d|a, d|b,$</span> and <span class="math-container">$d|c$</span>. I'm not sure what to do from here or if I'm even going in the righ... | Bill Dubuque | 242 | <p><span class="math-container">${\rm By\ Euclid}\ \ b\mid c\,\Rightarrow\, (a,b) = (a,b,c) = 1,\,$</span> by <span class="math-container">$\,(a,c) = 1$</span></p>
<p><span class="math-container">${\rm Equivalently\!\!:}\ \ \underbrace{d\mid a,b\,\Rightarrow\, d\mid a,b,c}_{\large d\ \mid\ b\ \mid\ c}\,\Rightarrow\, ... |
4,552,917 | <blockquote>
<p>Coin A is a fair coin and coin B is weighted so that the probability of obtaining heads is 0.6. An experiment consists of randomly choosing a [single, fixed] coin and then flipping it 10 times. Given that the coin came up heads exactly 6 times in the 10 flips, what is the probability that you selected... | math and physics forever | 879,009 | <p>One way to solve it is to note the function is odd about <span class="math-container">$x=1+{\pi}$</span>. Hence the integral(which is essential signed area) , is odd</p>
|
34,664 | <p>Here is the post for context: <a href="https://math.stackexchange.com/questions/4415677/is-today-is-presidents-day-a-proposition-propositional-function-or-neither#4415684">Is "today is Presidents' Day" a proposition, propositional function, or neither?</a></p>
<p>Both the answers are really great and t... | Caleb Stanford | 68,107 | <p>Various people have given various suggested criteria in the comments -- including accepting the first answer, the one posted by the lower-rep user, and other ideas. If both answers are truly "equally good", it doesn't matter -- pick any of these criteria, or choose your own, or even flip a coin!</p>
<p>Reg... |
34,664 | <p>Here is the post for context: <a href="https://math.stackexchange.com/questions/4415677/is-today-is-presidents-day-a-proposition-propositional-function-or-neither#4415684">Is "today is Presidents' Day" a proposition, propositional function, or neither?</a></p>
<p>Both the answers are really great and t... | John Bentin | 875 | <p>In reference to the particular answers you cite, I agree that they are both good. However, I would accept the first because it is shorter.</p>
<p>In general, when there is little to choose between answers as to content, base your acceptance on secondary criteria—in particular, style. Thus, consider brevity and clari... |
3,806,405 | <p>Find all (strictly) increasing functions <span class="math-container">$f$</span> from
the nonnegative integers to the integers satisfying <span class="math-container">$f(2) = 7$</span> and
<span class="math-container">$$f(mn) = f(m) + f(n) + f(m)f(n)$$</span>
for all nonnegative integers <span class="math-container"... | Oldboy | 401,277 | <p>If your number is <span class="math-container">$n$</span> and if you add <span class="math-container">$k$</span> zeroes after it, you get <span class="math-container">$a=n\cdot 10^k$</span>. We are going to look for a prime greater than <span class="math-container">$a$</span> and less than <span class="math-containe... |
3,806,405 | <p>Find all (strictly) increasing functions <span class="math-container">$f$</span> from
the nonnegative integers to the integers satisfying <span class="math-container">$f(2) = 7$</span> and
<span class="math-container">$$f(mn) = f(m) + f(n) + f(m)f(n)$$</span>
for all nonnegative integers <span class="math-container"... | Qiaochu Yuan | 232 | <p>It's a big hammer, but at least for the purposes of thinking through what sorts of results should and shouldn't be true of the primes, it's worth noting that this follows straightforwardly from the <a href="https://en.wikipedia.org/wiki/Prime_number_theorem" rel="nofollow noreferrer">prime number theorem</a> (PNT). ... |
1,218,259 | <p>Let $(M,g)$ be an Einstein manifold with Levi-Civita connection $\nabla$ and whose Ricci tensor $\text{Rc}(g)=g$, in components $R_{ij}=g_{ij}$. The <strong>Lichnerowicz Laplacian</strong> of $g$ is the map
\begin{align*}
\Delta_L:\Gamma(S^2M)&\rightarrow\Gamma(S^2M)\\
h&\mapsto(\Delta_Lh)_{ij}=g^{st}\nabla... | Mathemagician1234 | 7,012 | <p>What you're really asking for a textbook giving a modern and careful presentation of vector calculus/calculus of functions of several variables.Since you're reading Evan's text,I'll assume you've had a least a good undergraduate course in real analysis of one variable on metric spaces a la "baby Rudin" or Pugh. I'll... |
1,923,416 | <p>I would like to calculate the exponential map in the n-sphere, however, i don't know how to get started. Someone could give me a tip or bibliographic reference?</p>
| Andrew D. Hwang | 86,418 | <p>$\newcommand{\Vec}[1]{\mathbf{#1}}$If $\Vec{v}$ is a non-zero tangent vector to the $n$-sphere at a point $\Vec{p}$, the geodesic starting at $\Vec{p}$ with initial velocity $\Vec{v}$ is a circle of speed $\|\Vec{v}\|$ lying in the plane spanned by $\Vec{p}$ and $\Vec{v}$:
$$
\gamma(t) ... |
40,094 | <p>I have a table form as:</p>
<pre><code>table = {{a, 6}, {b, ff}, {c, 2}}
</code></pre>
<p>I want to associate the first element of each pair from the table to the second element of the pair as in:</p>
<pre><code>a=6, b=ff, c=2.
</code></pre>
<p>I want the parameters a,b,c to be recognized throughout the program... | Mr.Wizard | 121 | <p>This may be considered a duplicate of <a href="https://mathematica.stackexchange.com/q/6669/121">Assign the results from a Solve to variable(s)</a> which despite different formulation shares the same shortest answer:</p>
<pre><code>Set @@@ {{a, 6}, {b, ff}, {c, 2}}
</code></pre>
<p>This may also be related to <a h... |
104,463 | <p>For the Bézier surface, which owns the following <em>matrix definition</em>:</p>
<p>$$\begin{align*}
\mathbf S(u,v)&=\sum_{i=0}^m \sum_{j=0}^n \mathbf P_{i,j} B_{m,i}(u) B_{n,j}(v)\\
&=\small \begin{pmatrix}B_{m,0}(u)&\cdots&B_{m,m}(u)\end{pmatrix}_{1\times(m+1)}
\begin{pmatrix}
\mathbf P_{0,0}&... | xzczd | 1,871 | <h2>Update</h2>
<p>Fully compiling the code to C makes it as fast as the built-in:</p>
<pre><code>cBernstein =
Compile @@ (Hold[{n, {j, _Real, 1}, u}, Table[expr u^i (1 - u)^(n - i), {i, j}],
CompilationTarget -> C] /. expr -> FunctionExpand[Binomial[n, i]])
BezierSurface4 =
With[{cBernstein = cBernst... |
104,463 | <p>For the Bézier surface, which owns the following <em>matrix definition</em>:</p>
<p>$$\begin{align*}
\mathbf S(u,v)&=\sum_{i=0}^m \sum_{j=0}^n \mathbf P_{i,j} B_{m,i}(u) B_{n,j}(v)\\
&=\small \begin{pmatrix}B_{m,0}(u)&\cdots&B_{m,m}(u)\end{pmatrix}_{1\times(m+1)}
\begin{pmatrix}
\mathbf P_{0,0}&... | J. M.'s persistent exhaustion | 50 | <p>This might help you get an idea:</p>
<pre><code>n = 4;
BlockRandom[SeedRandom[42, Method -> "Legacy"]; (* for reproducibility *)
cpts = Table[{i, j, RandomReal[{-1, 1}]}, {i, n + 1}, {j, n + 1}]];
GraphicsRow[{ParametricPlot3D[BezierFunction[cpts][u, v], {u, 0, 1}, {v, 0, 1},
... |
1,194,527 | <p>In several Linear Alegebra textbooks there is always a property of vector spaces listed that seems pointless to mention given it can be simplified.
Can someone please tell me the significance of it that warrants it's mention.</p>
<p><strong>Property in question:</strong>
<em>If a, b are two numbers, then (a+b) <str... | Phoenix87 | 76,702 | <p>Consider the given representation as sitting inside the universal representation. Then there exists a projection $E$ in the commutant of the enveloping von Neumann algebra which projects onto the representation subspace. Now consider the family $F$ of central projections that contain $E$. This is clearly non-empty a... |
433,130 | <p>A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less traditional examples.</p>
<p>Let <span class="math-container">$A$</span> be a commutative ring, and <span class="math-... | David White | 11,540 | <p>An example that might help the OP (who has "tried to come up with less traditional examples") is spelled out in Bill Dwyer's book chapter <a href="https://link.springer.com/chapter/10.1007/978-94-007-0948-5_1" rel="nofollow noreferrer">Localizations</a> (Examples 2.5-2.7). This example does not exactly mat... |
794,268 | <p>Binomial theorem is a very popular theorem that:
$$(x + y) ^ n = \sum_{i=0}^n {n \choose i}x^i y^{n-i}$$</p>
<p>I am looking for any papers (the newer the better) where I can find any informations about asymptotic approximation of the binomial theorem, especially for the case when I have $(1 + y)^n$.
<a href="http... | Antonio Vargas | 5,531 | <p>It was mentioned in the comments that the author is seeking an approximation in the case that $n \to \infty$ and $y \approx 1/n$. One approach is to set $y = \frac{1+z}{n}$, so that the expression is</p>
<p>$$
\left(1 + \frac{1+z}{n}\right)^n.
$$</p>
<p>Then we can expand the function</p>
<p>$$
f(a,b) = (1 + (1+... |
3,673,915 | <p>If this is a duplicate you can close this post.</p>
<blockquote>
<p>Introduction to Real Analysis (Robert G. Bartle) 18. (b) </p>
<p>If <span class="math-container">$f$</span> is a bijection of <span class="math-container">$A$</span> onto <span class="math-container">$B$</span>, show that <span class="math-c... | Gibbs | 498,844 | <p>Showing that if <span class="math-container">$x \neq y$</span> then <span class="math-container">$f^{-1}(x)\neq f^{-1}(y)$</span> is equivalent to prove that if <span class="math-container">$f^{-1}(x) = f^{-1}(y)$</span> then <span class="math-container">$x=y$</span>. But if <span class="math-container">$f^{-1}(x) =... |
3,526,650 | <p>I was studying <code>Shannon</code>'s <code>Entropy</code> function and for a 35% chance of a particular event, the formula produced the answer 1.5 bits.</p>
<pre><code>log2(0.35) = 1.5 bits(approx.) of information.
</code></pre>
<p>I know it's pretty trivial in context of practical applications, but how can one v... | Zaid Khan | 269,425 | <p>Thank you to <a href="https://math.stackexchange.com/questions/3526650/how-can-i-think-of-visualize-0-5-bits-of-information#comment7253189_3526650">Jyrki Lahtonen</a> in the comments for answering it convincingly.</p>
<blockquote>
<p>You seem to be confusing the set of possible values of a bit (as
information s... |
2,158,115 | <p>I am trying to write down the set of all ordered pairs $(n,m)$ such that $n=2k+1:k\in\mathbb{Z}^{+}$ and $m=2l+1:l\in\mathbb{Z}^{+}$. How would that be written in set notation?</p>
| mvw | 86,776 | <p>One way is:
$$
\{ (2k+1, 2l+1) \mid k, l \in \mathbb{Z}^+ \}
$$</p>
|
2,356,789 | <p><strong>Suppose you have a set of x objects.</strong> In it is an object y. Suppose you pick at random one object from this set. This set is uniformly distributed. There is a 1 / x chance of picking y. Let's say that you pick x times from the set, then eliminating it from the set. This means there is a 100% chance o... | Ethan Bolker | 72,858 | <p>This is like flipping an unfair coin. The standard way to answer your question uses a trick: think about failing to pick the right object in each of $n$ independent draws (with replacement). That happens with probability
$$
\left(
1- \frac{1}{x}
\right)^n .
$$
The larger $n$ the nearer that is to $0$, so the chanc... |
1,345,008 | <p>In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example</p>
<blockquote>
<p>$$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$</p>
</blockquote>
<p>How would you use calculus, just a sketch, I am interested in the method itself. </p>
| Ian | 83,396 | <p>For this I'd suggest Lagrange multipliers. Using the usual terminology, your objective function is $f(a,b,c)=a^2b+b^2c+c^2a$ and your constraint function is $g(a,b,c)=a+b+c=3$. So you introduce the Lagrange multiplier variable $\lambda$ and solve the system of four equations</p>
<p>$$\nabla f = \lambda \nabla g \\
... |
745,922 | <p>How did Leibniz know to write derivatives as $$\frac{dy}{dx}$$ so that everything would work out? For example, the chain rule: $$\frac{dy}{dz}=\frac{dy}{dx}\frac{dx}{dz}$$ Integration by Parts: $$xy=\int d(xy)=\int x\,dy+y\,dx \implies \int x\,dy =xy-\int y\,dx$$ Separable differential equations: $$\frac{dy}{dx}=\fr... | littleO | 40,119 | <p>A derivative really is <em>approximately</em> a ratio of two extremely tiny (but not "infinitely small") numbers: $f'(x) \approx \frac{\Delta f}{\Delta x}$. So, many arguments using "infinitesimals" actually make sense, if you just think of $df$ and $dx$ as being extremely tiny numbers and use $\approx$ instead of ... |
2,431,679 | <p>First I denote $x=\arctan{2}$ and $y=\arcsin{\frac{1}{\sqrt{10}}}$ and then use the addition formula for sine:</p>
<p>$$\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}=\sin{x}\cos{y}-\cos{x}\cdot \frac{1}{\sqrt{10}}.$$</p>
<p>Now I use the fact that $\cos{y}=1-\sin^2{y}$ which gives</p>
<p>$$\cos{y}=1-\sin^2{\left(\arc... | drhab | 75,923 | <p>Hint:</p>
<p>$\frac{\sin x}{\cos x} =\tan(x)=2$ and $\sin^2x+\cos^2x=1$</p>
|
2,431,679 | <p>First I denote $x=\arctan{2}$ and $y=\arcsin{\frac{1}{\sqrt{10}}}$ and then use the addition formula for sine:</p>
<p>$$\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}=\sin{x}\cos{y}-\cos{x}\cdot \frac{1}{\sqrt{10}}.$$</p>
<p>Now I use the fact that $\cos{y}=1-\sin^2{y}$ which gives</p>
<p>$$\cos{y}=1-\sin^2{\left(\arc... | Guy Fsone | 385,707 | <p>Hint:</p>
<p>$$\sin x =\sqrt{1-\cos^2 x} =\sqrt{1-\frac{1}{1+\tan^2 x}} $$</p>
<p>$$\cos x =\sqrt{\frac{1}{1+\tan^2 x}} $$
Since </p>
<p>$$ 1+\tan^2 x =1+\frac{\sin x^2}{\cos^2 x} = \frac{\cos^2x+\sin x^2}{\cos^2 x} =\frac{1}{\cos^2 x}$$</p>
|
2,431,679 | <p>First I denote $x=\arctan{2}$ and $y=\arcsin{\frac{1}{\sqrt{10}}}$ and then use the addition formula for sine:</p>
<p>$$\sin{(x-y)}=\sin{x}\cos{y}-\cos{x}\sin{y}=\sin{x}\cos{y}-\cos{x}\cdot \frac{1}{\sqrt{10}}.$$</p>
<p>Now I use the fact that $\cos{y}=1-\sin^2{y}$ which gives</p>
<p>$$\cos{y}=1-\sin^2{\left(\arc... | Khosrotash | 104,171 | <p><a href="https://i.stack.imgur.com/vHKy1.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/vHKy1.jpg" alt="enter image description here"></a>$$\sin{(\arctan{2}-\arcsin{\frac{1}{\sqrt{10}}})} = \\\sin (x-y) =\\ \sin x \cos y- \cos x \sin y= \frac{2}{\sqrt 5}.\frac{3}{{\sqrt {10}}}-\frac{1}{\sqrt 5}.\... |
3,032,802 | <p>Does anyone know how to show <span class="math-container">$ \lim_{n\rightarrow\infty} \sum_{k=1}^{\infty}k^{1/2} n^{-1} (1 - \cos(n^2 k^{-2}))=\infty$</span>? My thoughts were that you should you a series approximation of some sort, but I don't know how to get through it.</p>
| marty cohen | 13,079 | <p>Let
<span class="math-container">$s(n, m)
= \sum_{k=1}^{m}k^{1/2} n^{-1} (1 - \cos(n^2 k^{-2}))
$</span></p>
<p><span class="math-container">$1 - \cos(n^2 k^{-2})
= 2\sin^2(\frac{n^2}{2k^2})
$</span>
so
<span class="math-container">$s(n,m)
= 2\sum_{k=1}^{m}k^{1/2} n^{-1}\sin^2(\frac{n^2}{2k^2})
$</span>.</p>
<p>If... |
2,143,658 | <blockquote>
<p>Show that no two of the spaces $(0, 1), (0, 1], [0, 1]$ are homeomorphic</p>
</blockquote>
<p><em>My Attempted Proof</em></p>
<p>We have $(0, 1) \subset (0, 1] \subset [0, 1]$.</p>
<p>We also have $(0, 1] = (0, 1) \cup \{1\}$ and thus $|(0, 1)| \neq |(0, 1]|$ and thus no bijective mapping $f : (0,... | Henno Brandsma | 4,280 | <p>Define the following topological properties, using the notion of a "cut point".
If $X$ is connected then $x$ is a <em>cutpoint</em> of $X$ iff $X \setminus \{x\}$ is disconnected. </p>
<p>It's clear that when $f$ is a homeomorphism between $X$ and $Y$, then $x$ is a cut point for $X$ iff $f(x)$ is a cut point for $... |
2,308,093 | <p>Let $u_0,u_1,u_2,\ldots$ be the sequence
$0, 1, 1/2,3/4, 5/8, 11/16,\ldots$
in which the average of the previous two terms forms the current term, i.e. $$u_{s+2} = \left( \frac{u_{s+1} + u_s}{2}\right)$$
Let $v_s = u_s- u_{s-1}$.</p>
<p>The conjectured formula for $v_n$ is $(-1/2)^{(n-1)}$, but to prove it by indu... | Mark Bennet | 2,906 | <p>You can start with $$2u_{n+2}=u_{n+1}+u_n$$ and subtract $2u_{n+1}$ from each side to obtain $$2u_{n+2}-2u_{n+1}=u_n-u_{n+1}$$ or $$u_{n+2}-u_{n+1}=-\frac 12\left(u_{n+1}-u_n\right)$$</p>
<p>From which it is easy to establish what you want.</p>
|
3,440,648 | <blockquote>
<p>Prove that <span class="math-container">$$ \lim_{x \to 0} \frac{ \sqrt{x + \sqrt{ x + \sqrt{x}}} }{\sqrt[8]{x}}= 1$$</span> </p>
</blockquote>
<p>Let <span class="math-container">$x=y^8$</span> with <span class="math-container">$t>0$</span>. Then after we get
<span class="math-container">$$
\lim ... | Brady Gilg | 188,927 | <p>Instead of representing <span class="math-container">$\frac{a}{b}$</span> as a fraction, represent it as the vector <span class="math-container">$\left( \begin{array}{c} a \\ b \end{array} \right)$</span>.</p>
<p>Then, all you are doing to generate your sequence is repeatedly multiplying by the matrix <span class="... |
3,440,648 | <blockquote>
<p>Prove that <span class="math-container">$$ \lim_{x \to 0} \frac{ \sqrt{x + \sqrt{ x + \sqrt{x}}} }{\sqrt[8]{x}}= 1$$</span> </p>
</blockquote>
<p>Let <span class="math-container">$x=y^8$</span> with <span class="math-container">$t>0$</span>. Then after we get
<span class="math-container">$$
\lim ... | Community | -1 | <p>You are re-stating the Fibonacci sequence, and by the theory of linear recurrences, the terms are quasi-proportional to the powers of the largest root of the characteristic equation </p>
<p><span class="math-container">$$\phi^2-\phi-1=0.$$</span></p>
<p>Hence, the ratio of successive terms quickly tends to <span c... |
313,064 | <p>I have a problem with an exercise:
Find the probability that when throwing three dice, the six will occur on one of them (no matter which one), if on the facets of the remaining two dice, two different numbers will outcome.</p>
<p>The answer given to this exercise is 1/2. However I think it is $\frac{60}{6^3}=\frac... | Asinomás | 33,907 | <p>number of cases total: $6^3$. number of cases in which there is a 6 and no two equal results: $1*5*4*3$. So the result is $\frac{60}{216}$ I agree with you.</p>
<p>If the 6 is in the first roll then there are 20 other combinations, if the 6 is in the second the same, and if it is in the third the same, you don't re... |
3,570,313 | <blockquote>
<p>Prove that there exist infinitely many pairs <span class="math-container">$(a,b)$</span> of relatively prime positive integers
such that <span class="math-container">$\frac{a^2-5}{b}, \frac{b^2-5}{a}$</span> are both positive integers.</p>
</blockquote>
<p>I saw that this problem came from Germany ... | Will Jagy | 10,400 | <p>This is from Batominovski's anwer, make this CW. Worth remembering</p>
<p>Suppose we have constant integers <span class="math-container">$V,W$</span> and nonzero integer variables <span class="math-container">$x,y$</span> with the requirement that we always have <span class="math-container">$\gcd(x,y) = 1.$</span... |
2,596,972 | <blockquote>
<p>Solve the Diophantine equation or prove there are no solutions. $$(8n)!-4n+1=k^2$$</p>
</blockquote>
<p>I try to use that $k^2 \equiv 0\text{ or }1 \pmod 3$ and $k^2 \equiv 0\text{ or }1 \pmod 4$ but I can not come to a contradiction. Thankful for any ideas.</p>
| aid78 | 511,281 | <p>The number $4n-1$ is not an exact square $4n-1\equiv 3 (mod 4)$. </p>
<p>Therefore, in the canonical decomposition, it has a factor of the form $p^k$, where $p$ is prime and $k$ is odd. The factorial has factors $4n-1$ and $8n-2$, therefore it is divisible by $p^{2k}$. </p>
<p>The difference is divided by $p^k$, ... |
1,230,128 | <p>I was reading an article that said: if I have a finite field, say $\mathbb{F}_q^k$ where $q=p^n$ and $p$ is a prime; and a (k-2)-dimensional subspace, say $U\subset \mathbb{F}_q^k$ given by the span of $k-2$ linearly independent vectors of $\mathbb{F}_q^k$, then there exists $q+1$ hyperplanes in $\mathbb{F}_q^k$ th... | Crostul | 160,300 | <p>Recall the bijection between $\{ \mbox{hyperplanes of $\Bbb{F}_q^k$ containing $U$} \}$ and $\{ \mbox{hyperplanes of $\Bbb{F}_q^k/U$} \}$.</p>
<p>In particular, $\Bbb{F}_q^k/U$ is a $2$-dimensional $\Bbb{F}_q$-vector space, hence hyperplanes of $\Bbb{F}_q^k/U$ are as many as lines (subspaces of dimension $1$) in $\... |
1,230,128 | <p>I was reading an article that said: if I have a finite field, say $\mathbb{F}_q^k$ where $q=p^n$ and $p$ is a prime; and a (k-2)-dimensional subspace, say $U\subset \mathbb{F}_q^k$ given by the span of $k-2$ linearly independent vectors of $\mathbb{F}_q^k$, then there exists $q+1$ hyperplanes in $\mathbb{F}_q^k$ th... | k.stm | 42,242 | <p><em>Hint</em>. By the projection $\mathbf F_q^k → \mathbf F_q^k/U$, the $(k-1)$-dimensional subspaces (i.e. hyperplanes) of $\mathbf F_q^k$ containing $U$ directly correspond to the $1$-dimensional subspaces of $\mathbf F_q^k/U \cong \mathbf F_q^2$.</p>
|
1,230,128 | <p>I was reading an article that said: if I have a finite field, say $\mathbb{F}_q^k$ where $q=p^n$ and $p$ is a prime; and a (k-2)-dimensional subspace, say $U\subset \mathbb{F}_q^k$ given by the span of $k-2$ linearly independent vectors of $\mathbb{F}_q^k$, then there exists $q+1$ hyperplanes in $\mathbb{F}_q^k$ th... | Timbuc | 118,527 | <p>If $\;\{u_1,...,u_{k-2}\}\;$ is a basis for $\;U\;$ , then any hyperplane that contains it will have a basis of the form $\;\{u_1,...,u_{k-2},v\}\;$ . We can take $\;v\;$ out from a two dimensional space (what is "left out" in $\;\Bbb F_q^k\;$ after taking out $\;U\;$), so we have $\;q^2-1\;$ different <em>non-zero<... |
2,890,432 | <p>Imagine a tree made up only of vertices of degree 1 and vertices of degree $n$.</p>
<p>Let $m$ be the number of vertices of degree 1 and $\frac{m-2}{n-2}$ be the number of $n$-vertices of the tree.</p>
<p>How can I find the number of vertices <em>not</em> connected to a vertex of degree 1? I.e. the number of $n$-v... | Mike | 544,150 | <p>You can't, at least not from the degree sequence alone. Consider a tree $T_1$ where node A is the root its neighbours are B,C,and D, and in turn B,C, and D have 2 leaf children each. Then $T_1$ has 4 vertices of degree 3--namely $A,B,C$ and $D$--and 6 vertices of degree 1, and exactly one vertex NOT connected to a v... |
568,930 | <p>Prove or disprove: There exists an element in the quotient group $\mathbb{R}/\mathbb{Z}$ of order 6. By counting formula, since $|\mathbb{R}/\mathbb{Z}|$ is infinite, it seems possible to have an element of order 6. But how to find one? In addition, what is $\mathbb{R}/\mathbb{Z}$, please?</p>
| Aaron | 9,863 | <p>Since the groups we are dealing with are abelian, I will use $+$ as the group operation. </p>
<p>The elements of $\mathbb R/\mathbb Z$ are cosets of the subgroup $\mathbb Z \subset \mathbb R$. For example, the identity element is the coset $[0]=0+\mathbb Z$, and $[.5]=.5 + \mathbb Z$ is another element. Note tha... |
151,516 | <p>I am trying to find this by using integration by parts but I am not sure how to do it.</p>
<p>$$\int_0^{\pi/2} (\sin x)^7 (\cos x)^5 \mathrm{d} x$$</p>
<p>I tried rewriting as </p>
<p>$$\int_0^{\pi/2} \sin x\cdot\ (\sin x)^6\cdot\ (\cos x)^5 \mathrm{d} x = \int_0^{\pi/2}\sin x(1-\ (\cos x)^3)\cdot\ (\cos x)^5 \m... | clathratus | 583,016 | <p>I'm going to expand on Valentin's answer:</p>
<p><span class="math-container">$$I(a,b)=\int_0^{\pi/2}\sin^ax\ \cos^bx\ dx$$</span>
<span class="math-container">$t=\sin^2x$</span>:</p>
<p><span class="math-container">$\therefore dt=2\sin x\cos x\ dx\\\therefore dx=\frac12t^{-1/2}(1-t)^{-1/2}dt\\\therefore x=0\mapst... |
3,670,963 | <p>I am trying to calculate the probability of taking from a regular deck of cards (52) 5 cards with 4 different suits, my go: </p>
<p>first we think about picking 'diamond' (just to make it more simple) it is: <span class="math-container">$\frac{13}{52}$</span><br>
second we think about picking 'heart': <span class=... | user | 293,846 | <p>Obviously, the "correct" hand should contain two cards of a suit and one card of each of the other 3 suits. There are <span class="math-container">$\binom41$</span> ways to choose the suit with the double, and
<span class="math-container">$\binom{13}2$</span> ways to choose the double out of the suit. Therefore the... |
177,181 | <p>I have a object which is similar to the shape of irregular rock like this <img src="https://i.stack.imgur.com/Zz55K.gif" alt="enter image description here"></p>
<p>I would like to find the volume of this. How to do it?</p>
<p>If I have to find the volume, what are the things I would need. eg., If it is cylindrical... | nbubis | 28,743 | <p>The easiest way (if you have a 3d model) is simple Monte Carlo integration. One builds a big bounding box around the "rock", and then randomly chooses a large number of points inside the box. The ratio of points inside the "rock" to the the number of total points, is the ratio of volumes. Since you can easily calcul... |
1,361,766 | <p>I am stucked at this combinatorics/probability problem:</p>
<hr>
<p>There are 10 urns, each contains 8 balls numbered 1,2,...,8</p>
<p>If we randomly choose 1 ball from <strong>each</strong> urn, what is the probability that the <strong>total</strong> number of 1's will be <strong>exactly</strong> 3 and the <stro... | true blue anil | 22,388 | <p>Using the multinomial distribution formula as there are <strong>3 categories (1's,4's and the rest)</strong></p>
<p>Pr = $\dfrac{10!}{3!2!5!} \times \left(\dfrac{1}{8}\right)^3\times\left(\dfrac{1}{8}\right)^2\times\left(\dfrac{6}{8}\right)^5$</p>
<p>Its parallel to the binomial distribution should be obvious.</p... |
2,611,860 | <p>I am stuck in this exercise. it asks me to:</p>
<p>Proof that the only ring homomorphism $\phi:\Bbb Z_{n} \rightarrow \Bbb Z$ is the trivial one $\phi(n)=0$</p>
<p>I believe that I have to show that the kernel $K=\{n \in R|\phi(n)=0_{s} \}$</p>
<p>But I am not sure how to start, any hints will be appreciated</p>
| egreg | 62,967 | <p>If $x\in\mathbb{Z}_n$, then $nx=0$, so also $\phi(nx)=0$.</p>
<p>For a homomorphism, $\phi(nx)=n\phi(x)$. Now use that $n>0$.</p>
|
411,753 | <p>Sorry, I mis-typed the question. The real question is that<br>
Every normal matrix is diagonalizable?<br>
and the answer is False.<br>
Can you give me a counter example?</p>
| Community | -1 | <p>Let
$$A=\left(\begin{matrix}1&i\\
i&-1\end{matrix}\right)$$
then $A$ is a symmetric complex matrix and its characteristic polynomial
$$\chi_A(x)=\det(A-xI)=x^2$$
then if $A$ is diagonalizable then it's similar to the zero matrix which is clearly false.</p>
<p><strong>Added</strong> A complex matrix $A$ is ... |
3,687,299 | <p>I have a stupid question about continuity in higher dimensions.</p>
<p>There are maps, for example, <span class="math-container">$f(x,y)=\frac{xy^2}{x^2+y^4}$</span>, when <span class="math-container">$(x,y)\neq (0,0)$</span> and <span class="math-container">$f(x,y)=(0,0)$</span> when <span class="math-container">$... | uniquesolution | 265,735 | <p>For the same reason that traveling from New York to Florida takes a few hours if you fly straight south, but takes a lot more if you choose to go first to Los Angeles.</p>
|
3,687,299 | <p>I have a stupid question about continuity in higher dimensions.</p>
<p>There are maps, for example, <span class="math-container">$f(x,y)=\frac{xy^2}{x^2+y^4}$</span>, when <span class="math-container">$(x,y)\neq (0,0)$</span> and <span class="math-container">$f(x,y)=(0,0)$</span> when <span class="math-container">$... | WHLin | 669,073 | <p>This has something to do with the <em>uniform continuity</em> around 0. Given a fixed <span class="math-container">$\epsilon > 0$</span>, if there exist an universal constant <span class="math-container">$\delta_u$</span> such that </p>
<p><span class="math-container">$$|(x,y)|<\delta_u \implies |f(x,y)|<\... |
1,231,829 | <blockquote>
<p>Find two solutions in $\mathbb{R}$ for the Riccati equation $$y'+y+y^2 = 2$$
Which satisfy the following initial conditions:</p>
<p>a) $y = b$ when $x=0$, $-2 \leq b < 1$</p>
<p>b) $y = b$ when $x=0$, $b \geq 1$ or $b < -2$</p>
</blockquote>
<p><strong>What I tried to do:</strong>
F... | Lutz Lehmann | 115,115 | <p>You did nothing wrong up to your last conclusion. $b=\frac{a+1}{a}$ is equivalent to $a=\frac{1}{b-1}$ and
$$
y(x)=1+\frac{3(b-1)}{e^{-3x}(2+b)-(b-1)}=\frac{e^{-3x}(2+b)+2(b-1)}{e^{-3x}(2+b)-(b-1)}
$$
so that for $b=1$ and $b=-2$ everything cancels to a constant solution.</p>
<hr>
<p>The only property that disting... |
1,936,712 | <p>Suppose I have a die with sides $1,2,3,4,5,5$. What is the probability of getting any order of $1,2,3,4,5$ in $5$ rolls?</p>
<p>Would I be right in saying you can't use combinations to try and calculate this because the probability of each outcome isn't equally likely?</p>
| BruceET | 221,800 | <p><strong>Comment:</strong></p>
<p>For verification, here is a simulation of a million 5-roll experiments,
which should approximate the probability in the original question
correct to two or three decimal places. The vector <code>pr</code> governs the
probabilities of faces labeled 1 through 5. On the $i$th roll,
<co... |
259,382 | <p>it seems first easy to me, but now i am tossing my head against wall not being able to solve the problem. i need to check for convergence of this sequence below. i dont know how to start although it seems to be very easy one</p>
<p>$\lim_{n \to \infty} i^{3n} = help = ?$ </p>
<p>i need help here. do i have to work... | Nameless | 28,087 | <p>Write $$a_n=i^{3n}=(i^{3})^{n}=(-i)^{n}=(-1)^ni^n$$
The last limit doesn't exist! </p>
<p>Take $k_n=4n$ and $m_n=4n+2$. Then $(a_{k_n})$ and $(a_{m_n})$ are both subsequences of $(a_n)$ but
$$a_{k_n}=(-1)^{4n}i^{4n}=1\cdot 1=1\to 1$$
while
$$a_{m_n}=(-1)^{4n+2}i^{4n+2}=1\cdot (-1)=-1\to -1$$
as $n\to +\infty$</p>
|
2,604,548 | <p>I am trying to solve some <strong>ratio tests</strong>. I came across to my notes and i can't figure out how is that equation simplified.
Well, how is $\frac{\left(2\left(n+1\right)\right)!}{\left(n+1\right)!}\cdot \frac{n!}{\left(2n\right)!}$ simplified to $\frac{\left(2n+1\right)\left(2n+2\right)}{n+1}$</p>
| Dylan Zammit | 400,332 | <p>Hint: Try to simplify this: $$\dfrac{(n+1)!}{n!}$$By using: $n!=1 \cdot 2 \cdot 3 \cdots n.$</p>
|
1,833,470 | <p>I am having a problem with this calculus problem:</p>
<blockquote>
<p>The rate of change of the number of coyotes $N(t)$ in a population is
directly proportional to $650−N(t)$, where $t$ is the time in years. That
is,</p>
<p>$dN/dt=k(650−N)$.</p>
<p>When $t=0$, the population is $270$, and when $t=2... | callculus42 | 144,421 | <p>The differential equation is </p>
<p>$\frac{dN}{dt}=k\cdot (650−N)$</p>
<p>Separating the variables</p>
<p>$\frac{dN}{ (650−N)}=k\cdot dt$</p>
<p>Integrating both sides</p>
<p>$\int \frac{dN}{ (650−N(t))}=k\cdot\int dt$</p>
<p>$-ln(650-N(t))=k\cdot t+C$</p>
<p>$ln(650-N(t))=-k\cdot t+C$</p>
<p>$650-N(t)=C\cd... |
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